Even in the purely methodical context
the Buslaev's and Grecchi's construction connecting the
asymptotically repulsive “wrong sign”
complex potential in (REF )
with the
asymptotically well-behaved
double-well potential in (REF )
is exceptional. Indeed,
both of the related partner Hamiltonians
possessing the discrete and
real bound-state spectra
are still just ordinary
differential operators of the second order.
In contrast,
the analogous would-be isospectral partners \(\mathfrak {h}^{(BB)}\) 
of all of the
Bender's and Boettcher's [1]} models of the form
\(H^{(BB)}=-d^2/dx^2+V^{(BB)}(x)\) 
have only been constructed approximatively [2]}, [3]}, [4]}. Moreover,
even this indicated that
their form (which,
as required, was self-adjoint in \({\cal L}\) )
was non-local and extremely complicated,
represented by
differential operators of infinite order.
After all, such an explicit construction
experience only reconfirmed the expectable
contrast
between a guaranteed simplicity (i.e., computational tractability) of
“input” \(H\)  and
a prohibitive
complexity of a generic “output” \(\mathfrak {h}\) .
